20220113, 03:12  #276  
6809 > 6502
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Aug 2003
101×103 Posts
2^{2}×3×877 Posts 
Quote:


20220113, 06:31  #277 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·1,693 Posts 
This is the smallest GFN primes and the smallest GRU primes in bases b<=64
Note 1: we do not include the case where the base of the GFNs is perfect odd power and the case where the base of the GRUs is either perfect power or of the form 4*m^4 with integer m, since such numbers have algebra factors and are composite for all n or are prime only for very small n, such families for bases 2<=b<=64 are: Code:
base GFN family GRU family 4 {1} 8 1{0}1 {1} 9 {1} 16 {1}, 1{5}, {C}D 25 {1} 27 {D}E {1} 32 1{0}1 {1} 36 {1} 49 {1} 64 1{0}1 {1}, 1{L}, 5{L}, 1{9}, {u}v Note 2: All GFN base b and all GRU base b are strongprobableprimes (primes and strong pseudoprimes) to base b, since they are overprobableprimes (primes and overpseudoprimes) to base b (references: https://oeis.org/A141232 http://arxiv.org/abs/0806.3412 http://arxiv.org/abs/0807.2332 http://arxiv.org/abs/1412.5226 https://cs.uwaterloo.ca/journals/JIS...shevelev19.pdf), and all overpseudoprimes are strong pseudoprimes to the same base b, all strong pseudoprimes are Euler–Jacobi pseudoprimes to the same base b, all Euler–Jacobi pseudoprimes are Euler pseudoprimes to the same base b, all Euler pseudoprimes are Fermat pseudoprimes to the same base b, so don't test with this base (see https://mersenneforum.org/showthread.php?t=10476&page=2, https://oeis.org/A171381, https://oeis.org/A028491, also see https://oeis.org/A210454, https://oeis.org/A210461, https://oeis.org/A216170, https://oeis.org/A217841, https://oeis.org/A243292, https://oeis.org/A217853, https://oeis.org/A293626, https://oeis.org/A210454/a210454.pdf, https://cs.uwaterloo.ca/journals/JIS...hamahata44.pdf, all generalized repunits in base b^2 with length p (where p is prime not dividing b*(b^21)) are Fermat pseudoprimes to base b, thus there are infinitely many pseudoprimes to every base b), note that there are also (but very few) numbers in the simple families which are neither GFN families nor GRU families, which are pseudoprimes, e.g. for the family {5}25 in base 8 (which have the smallest prime 555555555555525, corresponding to the secondlargest base 8 minimal prime (start with b+1)), a smaller number 525 is 341 in decimal, which is Fermat pseudoprime and Euler pseudoprime (although not strong pseudoprime, but there are many examples of strong pseudoprimes to base 2 and/or base 3, e.g. the smallest composite number which is strong pseudoprime to both base 2 and base 3 is 1373653 (see https://oeis.org/A072276), which has no proper subsequence which is prime > base (b) in bases b = 55, 58, 59, 65, 66, 70, 79, 82, 95, 103, 112, 113, 115, 116, 117, 121, 127, 130, 133, 134, 135, 136, 137, 138, 139, 141, 146, 147, 149, 151, 152, 155, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 177, 179, 183, 184, 185, 187, 188, 189, 191, 192, 193, 195, 196, 197, 199, 200, ..., and the secondsmallest composite number which is strong pseudoprime to both base 2 and base 3 is 1530787, which has no proper subsequence which is prime > base (b) in bases b = 77, 91, 95, 98, 109, 113, 120, 123, 125, 127, 129, 131, 132, 135, 136, 139, 141, 142, 143, 144, 147, 151, 155, 159, 160, 161, 162, 169, 170, 173, 176, 177, 179, 181, 183, 184, 187, 188, 189, 190, 191, 192, 194, 197, 199, 200, ..., and if we assume a number which has passed the Miller–Rabin primality tests to both base 2 and base 3 is in fact prime, our data will be wrong for these bases b) to base 2 (and thus to base 8, since pseudoprimes to base b are always (the same type) pseudoprimes to base b^r for all r>1, and 8=2^3). Last fiddled with by sweety439 on 20220506 at 14:57 
20220113, 06:36  #278 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·1,693 Posts 
but it is my newer researching result, and I think that it is important, sometimes I update my old posts, such as #208 and #215, also there are posts which are lists of references: #140 and #154
Last fiddled with by sweety439 on 20220113 at 06:37 
20220113, 11:42  #279  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×1,693 Posts 
Quote:
the original text file for base 35 is too large (1743 KB) to upload, thus zipped it Last fiddled with by sweety439 on 20220113 at 11:43 

20220115, 07:11  #280 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×1,693 Posts 
Conjecture: There is no base b such that the largest minimal prime (start with b+1) and the secondlargest minimal prime (start with b+1) have the same number of digits in base b, note that in the original minimal prime (i.e. prime > base is not required), the three largest minimal primes in decimal (base 10) all have the same number of digits (60000049, 66000049, 66600049, all have 8 digits), and in base 2 the largest (and the only) two minimal primes are 10 and 11, both have 2 digits, also, in base 5, the largest two minimal primes are 14444 and 44441, both have 5 digits.
For the problem in this project (i.e. the minimal primes (start with b+1)), the largest and the secondlargest minimal primes (start with b+1) have the numbers of digits: (combine with the thirdlargest and the fourthlargest minimal primes (start with b+1), see the table below): Code:
base 1st largest 2nd largest 3rd largest 4th largest 2 2 N/A N/A N/A 3 3 2 2 N/A 4 3 2 2 2 5 96 6 5 5 6 5 4 4 2 7 17 10 8 7 8 221 15 13 11 9 1161 689 331 38 (conjectured) 10 31 12 8 8 12 42 30 9 8 Another conjecture: For any number n>=2, there exists a minimal primes (start with b+1) with exactly n digits in base b, for every enough large b Clearly, all 2digit primes (except "10" (i.e. = b) when b itself is prime) are minimal primes (start with b+1) base b, I conjectured that all bases b != 2, 6 have a 3digit minimal prime (start with b+1), also all bases b>4 have a 4digit minimal primes (start with b+1), all bases b>4 have a 5digit minimal primes (start with b+1), etc. (note that in the original minimal prime (i.e. prime > base is not required), all singledigit primes are minimal primes, and I conjectured that all bases b != 8 have a 2digit minimal prime, all bases b != 2, 4, 6, 7 have a 3digit minimal prime, all bases b != 2, 3, 4, 5, 7 have a 4digit minimal prime, all bases b != 2, 3, 4, 9 have a 5digit minimal prime, etc. (the bases with no ndigit minimal prime for given n is more complex, thus the problem in this project (i.e. the minimal primes (start with b+1)) is really better)) (for more data, see post 145) we can research: * the possible length of the minimal primes (start with b+1) * the possible (first,last) combo of the minimal primes (start with b+1) * for these minimal primes (start with b+1), the digit which appears the most times in this minimal prime (start with b+1) * the length such that there are the most minimal primes (start with b+1) * the (first,last) combo such that there are the most minimal primes (start with b+1) Last fiddled with by sweety439 on 20220115 at 15:54 
20220117, 06:17  #281 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×1,693 Posts 
find the set of the minimal primes (start with b+1) base b for various bases b (2<=b<=36) is the target of the project in this thread.
minimal prime (start with b+1) base b is always minimal prime (start with b'+1) base b' = b^n, if it is > b', for any integer n>1 original minimal prime (i.e. prime > b is not required) base b is always minimal prime (start with b+1) base b, if it is > b 
20220121, 15:44  #282 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×1,693 Posts 
zipped file for the minimal primes (start with b+1) in bases 17<=b<=36

20220121, 15:52  #283  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×1,693 Posts 
Quote:
* If n is an integer which is > 10, and the base 10 representation of n contains none of {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequences, prove that n is composite. Of course, this can be generalized to other bases, such as: * If n is an integer which is > 7, and the base 7 representation of n contains none of {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} as subsequences, prove that n is composite. * If n is an integer which is > 8, and the base 8 representation of n contains none of {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} as subsequences, prove that n is composite. * If n is an integer which is > 12, and the base 12 representation of n contains none of {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077} as subsequences, prove that n is composite. etc. Last fiddled with by sweety439 on 20220124 at 12:42 

20220124, 13:06  #284 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×1,693 Posts 
Thus, the total proof of base 10 includes these proofs:
* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes. (of course, they are > 10, thus this part (i.e. all these numbers are > 10) needs no proof) (we can use ECPP (such as Primo) to prove that the largest two numbers are defined primes (i.e. not merely PRPs), in this case of base 10, the largest number has only 31 digits and can be proved primality in <1 second, but in other case, such as base 13, 14, and 16, there are numbers > 10^10000 in the set, thus ECPP (or N1, N+1, if this prime 1 or +1 can be trivially factored, such as the case of base 14, the largest prime 5*14^196981 in this set) is need to prove their primality) * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. (this is the easiest part of all these parts, as we can use either trial division or Fermat test to prove their compositeness (if these numbers have small prime divisors, or these numbers fails the Fermat primality tests, then they are defined composite), unless the numbers are Fermat pseudoprimes to many bases (say bases 2, 3, 5, 7, 11) with no small divisors (say < 2^32), in this case, we need to run either Miller–Rabin primality test or Lucas primality test to prove their compositeness) * Prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite. (for this part, we use either covering congruences or algebraic factorization (or combine of them, such as the base 12 family {B}9B and the base 14 family 8{D}) to prove that all numbers in a given family (may be nonsimple family, such as many families in base 29 and 41) are composite) Last fiddled with by sweety439 on 20220124 at 20:26 
20220125, 00:24  #285 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·1,693 Posts 
A conjecture between this minimal prime problem (i.e. start with b+1) and the original minimal prime problem (i.e. p > b is not required): In all bases other than 2, 3, 6, the largest minimal prime (start with b+1) is not minimal prime when p > b is not required, equivalently, the largest minimal prime (start with b+1) always contain either at least one prime digit or (contain the string "10" as subsequence and the base (b) is prime), also equivalently, the largest minimal prime (start with b+1) is not equal (must be larger than) the largest minimal prime when p > b is not required.
For the bases such that the original minimal prime problem (i.e. p > b is not required) is solved, this conjecture is verified in these bases b: * b=5: this new minimal prime problem is also solved, and 1(0^93)13 contain the prime digit 3 and contain the string "10" as subsequence and the base (5) is prime * b=7: this new minimal prime problem is also solved, and (3^16)1 contain the prime digit 3 * b=8: this new minimal prime problem is also solved, and (4^220)7 contain the prime digit 7 * b=9: there is minimal prime (start with b+1) > the largest original minimal prime, such as 3(0^1158)11 * b=10: this new minimal prime problem is also solved, and 5(0^28)27 contain the prime digits 2, 5, 7 * b=11: there is minimal prime (start with b+1) > the largest original minimal prime, such as 55(7^1011) * b=12: this new minimal prime problem is also solved, and 4(0^39)77 contain the prime digit 7 * b=13: not verified (may be false), since there is no known minimal prime (start with b+1) which is > the largest original minimal prime * b=14: there is minimal prime (start with b+1) > the largest original minimal prime, such as 4(D^19698) * b=15: there is minimal prime (start with b+1) > the largest original minimal prime, such as (7^155)97 * b=16: there is minimal prime (start with b+1) > the largest original minimal prime, such as 5B(C^3700)D (D(B^32234) is only probable prime, not proven prime) * b=17: original minimal prime problem not solved (there is an unsolved family F1{9}), but there are two known minimal primes (start with b+1) > the largest known original minimal prime (i.e. 4(9^111333)): 97(0^166047)1 and F7(0^186767)1 * b=18: there is minimal prime (start with b+1) > the largest original minimal prime, such as 8(0^298)B * b=20: there is minimal prime (start with b+1) > the largest original minimal prime, such as C(D^2449) * b=22: (I do not know whether there is minimal prime (start with b+1) > the largest original minimal prime or not, but it is very likely) * b=24: there is minimal prime (start with b+1) > the largest original minimal prime, such as 2(0^313)7 * b=30: there is minimal prime (start with b+1) > the largest original minimal prime, such as O(T^34205) * b=42: there is minimal prime (start with b+1) > the largest original minimal prime, such as 2(f^2523) * b=60: (original minimal prime problem seems to be solved, and there is an unsolved family Z{x} in the minimal prime (start with b+1) problem) Last fiddled with by sweety439 on 20220125 at 23:41 
20220128, 23:07  #286  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·1,693 Posts 
Quote:
* b^n+1 (self dual) (b == 0 mod 2) * (b^n+1)/2 (self dual) (b == 1 mod 2) * (b^n1)/(b1) (self dual) * 2*b^n+1 and b^n+2 (b == 3, 5 mod 6) * 2*b^n+1 and (b/2)*b^n+1 (b == 0, 2 mod 6) * (2*b^n+1)/3 and (b^n+2)/3 (b == 1 mod 6) * (2*b^n+1)/3 and ((b/2)*b^n+1)/3 (b == 4 mod 6) * 2*b^n1 and b^n2 (b == 1 mod 2) * 2*b^n1 and (b/2)*b^n1 (b == 0 mod 2) * 3*b^n+1 and b^n+3 (b == 2, 4 mod 6) * 3*b^n+1 and (b/3)*b^n+1 (b == 0 mod 6) * (3*b^n+1)/2 and (b^n+3)/2 (b == 7, 11 mod 12) * (3*b^n+1)/2 and ((b/3)*b^n+1)/2 (b == 3 mod 12) * (3*b^n+1)/4 and (b^n+3)/4 (b == 1, 5 mod 12) * (3*b^n+1)/4 and ((b/3)*b^n+1)/4 (b == 9 mod 12) * 3*b^n1 and b^n3 (b == 2, 4 mod 6) * 3*b^n1 and (b/3)*b^n1 (b == 0 mod 6) * (3*b^n1)/2 and (b^n3)/2 (b == 1, 5 mod 6) * (3*b^n1)/2 and ((b/3)*b^n1)/2 (b == 3 mod 6) * (b1)*b^n1 and b^n(b1) * (b1)*b^n+1 and b^n+(b1) * (b+1)*b^n1 and b^n(b+1) * (b+1)*b^n+1 and b^n+(b+1) (b == 0, 2 mod 3) * ((b+1)*b^n+1)/3 and (b^n+(b+1))/3 (b == 1 mod 3) * ((b2)*b^n+1)/(b1) and (b^n+(b2))/(b1) (b == 1 mod 2) * ((b2)*b^n+1)/(b1) and ((1/A006519(b2))*b^n+A000265(b2))/(b1) (b == 0 mod 2) * ((2*b1)*b^n1)/(b1) and (b^n(2*b1))/(b1) Note: 1/A006519(b2) is not integer (for b == 0 mod 2), we should start with the smallest n such that (1/A006519(b2))*b^n is integer Last fiddled with by VBCurtis on 20220129 at 01:10 Reason: stop quoting yourself 

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